What is the expression for the non-trivial sum of x^x from 0 to 1?

  • Thread starter Jakim
  • Start date
  • Tags
    Sum
In summary, the conversation discusses evaluating a definite integral and the discovery of a well-known sum known as the Sophomore's dream. The individual also mentions attempting to find an expression for the sum in terms of the Meijer G-function. References to further reading on the topic are also mentioned.
  • #1
Jakim
5
0
Hi. I tried to evaluate a definite integral of [itex]x^x[/itex] from [itex]0[/itex] to [itex]1[/itex] and I have reached following sum:

[tex]\int_0^1 x^x = \sum_{n=1}^{\infty} \frac{(-n)^{1-n}}{n}[/tex]

Is there an expression of this sum, for example in terms of Meijer G-function? I tried to find [itex]x^x[/itex] as G-function form to integrate it but unsuccessful.

Thanks in advance.
 
Physics news on Phys.org
  • #2
I'm not sure whether it's related to the G-function or not, but the sum itself is well-known, and is called the Sophomore's dream.
 
  • #3
Thanks, I didn't know it has a name; I will look for articles about it. Anyway I have reached the sum the same way as it is shown in your link.
 
  • #5
Hi, I've downloaded it even :). It was an answer to some of my questions. Thanks.
 

FAQ: What is the expression for the non-trivial sum of x^x from 0 to 1?

What is a non trivial sum?

A non trivial sum is a mathematical term that refers to the sum of two or more numbers or variables that results in a value other than zero. In other words, it is a sum that is not immediately obvious or easily predictable.

Why is finding non trivial sums important?

Finding non trivial sums is important because it allows us to explore and understand complex mathematical relationships. It also helps us to solve problems and make predictions in various fields such as physics, engineering, and economics.

How do you find non trivial sums?

To find non trivial sums, you can use mathematical operations such as addition, subtraction, multiplication, and division. You can also use algebraic equations and formulas to manipulate and combine numbers and variables to create non trivial sums.

What are some examples of non trivial sums?

Some examples of non trivial sums include 3 + 5, which equals 8, and 2x + 3y, where x and y are variables, which results in a non trivial sum of 5.

What are the applications of non trivial sums?

Non trivial sums have various applications in different fields. For example, in physics, they can be used to calculate the total energy of a system. In economics, they can be used to model and predict market trends. In computer science, they can be used to analyze algorithms and optimize performance.

Similar threads

A Summing over continuum and uncountable numerocities
Replies
1
Views
2K
I Express the limit as a definite integral
Replies
16
Views
3K
A Multiplying divergent integrals using Hardy fields approach
Replies
3
Views
2K
I What is the difference between these two power series theorems?
Replies
5
Views
921
B I don't recognize this limit of Riemann sum
Replies
2
Views
2K
I On convolution theorem of Laplace transform: Schiff
Replies
4
Views
2K
MHB Evaluate the surface integral $\iint\limits_{\sum}f\cdot d\sigma$
Replies
1
Views
2K
MHB Can the Series Sum Be Expressed as an Integral as N Approaches Infinity?
Replies
3
Views
1K
I Multivariable fundamental calculus theorem in Wald
Replies
2
Views
2K
I Resolve the Recursion of Dickson polynomials
Replies
1
Views
2K

Hot Threads

Recent Insights

Back
Top